# Ant on a board

### Ant on a board

An ant is trapped in an 6x4 board with squares (looking at the shape of the board, we have 6 squares horizontally and 4 vertically).
Starting from the bottom left square and moving only horizontally and vertically through the squares (not diagonally), is it possible that it reaches the top right square so that it is set free?
Note that the ant must pass from every single square and only ONCE!
If it is possible, explain the pattern, otherwise, explain why it cannot be done.

(no matter how hard I've tried, I haven't made it, so, by intuition only, I assume it cannot be done - but I need an explanation).

Thank you

jason

Posts: 13
Joined: Wed Jan 11, 2017 10:17 am
Reputation: 1

### Re: Ant in a board

It is not possible. This is a chessboard colouring problem.

Colour the squares with a chessboard colouring (i.e. alternating black and white). Suppose the colour the ant starts on is white. As it moves the colours of the squares it vists are necessarily white, black, white, black, etc (because the only squares you can ever move to are the opposite colour of the square you are on). After visiting all 24 squares (or any even number of squares) it will be on a black square (regardless of the path it took), this can't be the top right square because it is coloured white.

Hope this helped,

R. Baber.

Guest

### Re: Ant in a board

Many thanks! Very much appreciated!

jason

Posts: 13
Joined: Wed Jan 11, 2017 10:17 am
Reputation: 1